[PDF] Drill problems on derivatives and antiderivatives - Arizona Math

14228_2drill.pdf

Drill problems on derivatives and

antiderivatives

1 Derivatives

Find the derivative of each of the following functions (wherever it is dened):

1.f(t) =t2+t31t4

Answer:f0(t) =2t31t2+4t5

2.y=13px+14

Answer:

dydx=16xpx

3.f(t) = 2t34t2+ 3t1. Also ndf00(t):

Answer:f0(t) = 6t28t+ 3; f00(t) = 12t8

4.y=px12

x

Answer:

dydx=12px+ ln(2)12 x

5.f(z) = ln(3)z2+ ln(4)ez

Answer:f0(z) = 2ln(3)z+ ln(4)ez

6.y=x2+ (2)x

Answer:

dydx=2x21+ [ln(2)](2)x

7.f() = 4p

Answer:f0() = ln(4)12p4p

1

8.f(x) =x2px3x

Answer:f0(x) =

2x12px!

3 x+x2pxln(3)3x

9.f(z) =3z25z2+ 7z

Answer:f0(z) =21(5z+ 7)2

10.f(w) = (5w2+ 3)ew2

Answer:f0(w) = 10wew2+ (5w2+ 3)2wew2= (5w2+ 8)2wew2

11.f(y) =eey2

Answer:f0(y) = 2yey2eey2

12.f(z) =pz(ez+ 1)2

Answer:f0(z) =12pz1(ez+ 1)22pzez(ez+ 1)3

13.w(t) = (t2+ 3t)(1e2t)

Answer:w0(t) = 2t+ 3 +e2t(2t2+ 4t3)

14.f(x) =q1cos(x)

Answer:f0(x) =sin(x)2q1cos(x)

15.f(y) =esin(y)

Answer:f0(y) = cos(y)esin(y)

16.z=qsin(t)

Answer:

dzdt=cos(t)2qsin(t)

17.f() =2sin() + 2cos()2sin()

Answer:f0() =2cos()

2

18.z= tan(e3)

Answer:

dzd=3sec2(e3)e3

19.y=esin(2)

Answer:

dyd=esin(2) + 2ecos(2)

20.f(y) = arcsin(y2)

Answer:f0(y) =2yp1y4

21.f() = ln(cos())

Answer:f0() =tan()

22.f(t) = ln(ln(t)) + ln(ln(2))

Answer:f0(t) =1tln(t)

23.g(t) = arctan(3t4)

Answer:g0(t) =31 + (3t4)2

24.f(z) =1ln(z)

Answer:f0(z) =1z(ln(z))2

25.f(t) = 2tet1pt

Answer:f0(t) = 2et+ 2tet+12tpt

2 antiderivatives

Find the denite and indenite integrals below:

1. Z 3t2t2 dt

Answer:3ln(jtj) +2t+C

3 2.Z 

3cos( ) + 3q 

d

Answer:3sin( ) + 2 q +C

3. Z x2+x+ 1x! dx

Answer:

12x2+x+ ln(jxj) +C

4. Z (3cos(x)7sin(x))dx

Answer:3sin(x) + 7cos(x) +C

5.

Z1cos2(x)dx

Answer:tan(x) +C

6. Z e sin(x)cos(x)dx

Answer:esin(x)+C

7. Z =4

0(sin(t) + cos(t))dt

Answer:1

8. Z sin()(cos() + 5)7d

Answer:

18(cos() + 5)8+C 9.

Z1p4xdx

Answer:2p4x+C

10. Z xe x2dx

Answer:12ex2+C

11.

Zxcos(x2)qsin(x2)dx

Answer:

qsin(x2) +C 4

12.Zexexex+exdx

Answer:ln ex+ex2!

+C 13.

Z[ln(z)]2zdz

Answer:

13[ln(z)]3+C

14. Z 1

01x2+ 2x+ 1dx

Answer:

12 15. Z 0 22x+ 4x2+ 4x+ 5dx

Answer:ln(5)

16. Z tsin(t)dt

Answer:tcos(t) + sin(t) +C

17. Z yqy+ 3dy

Answer:

23(y+ 3)qy+ 335y65

+C 18. Z t

2e5tdt

Answer:

 t

22t5+225

e5t5+C 19. Z 1

0arctan(y)dy

Answer:

4ln(2)2 20.

Zxp1x2dx

Answer:p1x2+C

5